78 LECTURES AND ESSAYS [1869- 



presumption in favour of the hypothesis that we have 

 some understanding of the matter in hand. But instead 

 of dwelling on these purely personal, and therefore in 

 different, matters let me signalise the important conse 

 quences for this whole controversy of the admission so 

 gracefully made by Dr. Stirling. In one word if Newton 

 is right in this matter it appears, on Dr. Stirling s and 

 Hegel s own showing, that the whole theory of the calculus, 

 which can be deduced from this one determination by the 

 very processes which Hegel himself calls mechanical (as 

 quoted above, p. 3), is &quot; really unimpeachable,&quot; is free 

 from the paradoxes enumerated in The Secret of Hegel 

 (above, p. i), and, in brief, is a reasoned system of mathe 

 matical truth which gives no ground whatever for Hegel s 

 assertion &quot; that there is a contradiction in the very 

 method on which, as a science, it rests &quot; (Hegel, iii. 274 ; 

 Secret, ii. 339). And with this result the mathematicians 

 may very well rest satisfied, and, while heartily inviting 

 metaphysicians to study so invaluable a branch of science, 

 and to find, if they can, its place in the whole of human 

 thought, they are quite entitled to use their own freedom 

 in deciding whether they shall concern themselves further 

 with the lucubrations of a philosopher who, as Dr. Stirling 

 himself suggests (p. 129), did not give himself the trouble 

 to examine Newton s investigations at first hand. 



But while this position is very legitimate to mathe 

 maticians as a class, I, personally, am bound to go further, 

 having undertaken in my former paper not merely to 

 vindicate Newton, but to inquire into the positive mathe 

 matical worth of the remarks which Hegel offers in order 

 to clear up in his own way that obscurity which he finds 

 spread over the statements of mathematicians on the 

 theory of the calculus. Now there are, of course, unwise 

 mathematicians, as there are unwise metaphysicians. It 

 was, therefore, quite possible for Hegel to find utterances 

 about the calculus in which an acute, even though mathe 

 matically unpractised, eye could detect obvious absurdity. 



